Pipe flow and losses

Why it matters

Real fluid rubs against pipe walls and churns through fittings, and that friction turns useful pressure into heat. The lost energy, called head loss, decides how big a pump must be, how far water can be sent through a main, and how much energy a building's chilled-water loop uses every year. Getting it wrong means pumps that cannot deliver, or pipes that are needlessly large and expensive.

Laminar or turbulent: the Reynolds number

The Reynolds number compares inertia to viscous forces:

Re = ρ V D / μ        (or V D / ν, with ν = μ/ρ)

• ρ density (kg/m³), V average velocity (m/s), D pipe inner diameter (m), μ dynamic viscosity (Pa·s).
• Water at 20 °C: ρ ≈ 998 kg/m³, μ ≈ 1.002 × 10⁻³ Pa·s.

For pipe flow, roughly:

• Re < 2,300: laminar, smooth layered flow.
• 2,300 < Re < 4,000: transitional, unpredictable.
• Re > 4,000: turbulent, the usual case for water in building and industrial pipes.

The Darcy–Weisbach equation

Head loss due to friction in a straight pipe (the "major" loss) is:

h_f = f · (L / D) · V² / (2 g)

• h_f is in metres of the flowing fluid.
• L is pipe length, D inner diameter.
• f is the dimensionless Darcy friction factor.

Convert head loss to a pressure drop with ΔP = ρ g h_f.

Finding the friction factor

• Laminar: f = 64 / Re, exactly.
• Turbulent: f depends on Re and the relative roughness ε/D. The classic tools are the Moody chart and the implicit Colebrook equation. A widely used explicit approximation, accurate to within a few per cent over most of the turbulent range, is Swamee–Jain:

f = 0.25 / [ log10( ε/(3.7 D) + 5.74 / Re^0.9 ) ]²

Typical roughness ε: drawn tubing (copper, plastic) ≈ 0.0015 mm; commercial steel ≈ 0.045 mm; cast iron ≈ 0.26 mm. Old pipes get rougher with corrosion and scale.

Be careful with the Fanning friction factor, used in some chemical engineering texts: it is one quarter of the Darcy factor.

Worked example

Water at 20 °C flows at V = 2 m/s through 100 m of 50 mm commercial steel pipe (ε = 0.045 mm).

Re   = 998 × 2 × 0.05 / 1.002e-3 ≈ 99,600       → turbulent
ε/D  = 0.045 / 50 = 0.0009
f    = 0.25 / [log10(0.0009/3.7 + 5.74/99,600^0.9)]² ≈ 0.0220
h_f  = 0.0220 × (100 / 0.05) × 2² / (2 × 9.81) ≈ 8.97 m
ΔP   = 998 × 9.81 × 8.97 ≈ 87.8 kPa

Almost 9 m of head is lost in 100 m. That is a lot: designers usually keep water velocities in this size of pipe lower (often around 1–2 m/s), partly to limit losses and noise. Because h_f grows with V², halving the velocity cuts the friction loss to roughly a quarter.

Minor losses

Valves, elbows, tees, entrances and exits add minor losses, written with a loss coefficient K:

h_m = K · V² / (2 g)

Typical K values: a sharp pipe entrance about 0.5, a standard 90° elbow about 0.3 to 1.5 depending on type, a fully open gate valve about 0.2, an exit into a tank 1.0. Take them from manufacturer data or a reference table. In a short system with many fittings, "minor" losses can be larger than the pipe friction.

The extended energy equation

Putting it together between points 1 and 2, with a pump adding head h_p:

P1/(ρg) + V1²/(2g) + z1 + h_p = P2/(ρg) + V2²/(2g) + z2 + h_f + Σh_m

This is the equation behind every pump-selection problem.

Common mistakes

• Using the Fanning factor in Darcy's equation (an error of four times).
• Using nominal pipe size as the diameter. Use the actual inner diameter for the pipe schedule.
• Getting viscosity units wrong. Centipoise (cP) is 10⁻³ Pa·s.
• Forgetting that f is not a constant. It changes with flow rate through Re.

Recap

• Re < 2,300 laminar, > 4,000 turbulent.
• Darcy–Weisbach: h_f = f (L/D) V²/(2g); laminar f = 64/Re; turbulent from Moody, Colebrook or Swamee–Jain.
• Minor losses: h_m = K V²/(2g).
• Real piping is designed to codes (plumbing, process piping, fire protection) and reviewed by licensed engineers; these methods are for learning and estimates.

Try it

On the Do tab, write the Darcy–Weisbach equation in Python and estimate the head loss in the worked-example pipe.